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# A ‘polymath’ project on the Wilcoxon test?

`Polymath’ is a set of projects in massive collaborative proof of mathematical results; Terry Tao and Timothy Gowers are two of the famous mathematicians involved.  There’s a new potential project  on Gowers’s blog, which he describes a being related to intransitive dice. As you know, if you read this blog, (a) I prefer the term non-transitive, and (b) this means it’s about the Wilcoxon test.

The idea of the conjecture is that you define an $$n$$-sided die by sampling uniformly with replacement $$n$$ numbers from $$1, 2,3,\dots,n$$ as the numbers on the sides, with the constraint that the numbers have to add up to $$n(n+1)/2$$. Rolling such a die $$M$$ times samples $$M$$ observations from a distribution on $$1,2,3,\\dots, n$$ with mean $$(n+1)/2$$.  You construct three such dice, $$A$$, $$B$$, and $$C$$.  Their distributions have the same mean, so the $$t$$-test would have no ability to distinguish data from the three distributions, no matter how large $$M$$ was. But the Wilcoxon test probably would. It’s assumed, but not yet proved, that the probability of a tie according to the Wilcoxon test goes to zero as $$n\to\infty$$.

The interesting conjecture is that if you see $$A$$ beat $$B$$ and $$B$$ beat $$C$$ according to the Wilcoxon test, the probability that $$A$$ beats $$C$$ goes to 1/2 as $$n$$ goes to infinity.   That is, given equal means, the Wilcoxon test is basically as non-transitive as possible.